Unit III, Focus on Mathematical Models. In Unit I, Witchcraft in Salem Village, we were trying
to understand the past, in Unit II, Earnings and Discrimination,
the present, and in this unit, Unit III, Population and Resources,
we are interested in predicting the future. We are interested
in questions, such as: 'What will population levels be like 20
years from now?' 'Will resources keep up?' 'Can we make a better
future by our actions today?' Such are the questions, and we'll
try to answer them by using mathematical models.

Examples and Objectives of Mathematical
Models. What do you think of when
you hear the words, "Mathematical Modeling?" What are
some examples of mathematical models? Click below for responses
of students:

General weather forecasting, global warming,
flight simulation, hurricane forecasting, nuclear winter, nuclear
arms race, ... might come to mind as examples of large mathematical
models with a large potential impact on us all. Mathematical models
are also used to describe traffic flow, stock market options,
predator-prey relations, and techniques of search.

What are the objectives of mathematical modeling?
Forecasting the future, preventing an unwanted future, and understanding
various 'natural' and unnatural phenomena are some possibilities
expressed in very general terms. These might all be put into the
category of problem solving by using mathematics to mirror
an aspect of the world.

Example of Model

Objective

Overarching Objective

Weather

Prediction

Explanation

and

Understanding

Flight
Simulation

Training

Nuclear
Arms Race

Strategy
Development

Traffic
Flow

Regulation

Predator-Prey

Management

It's important not to confuse the mirror with what it's mirroring.
As the linguist S.I. Hayakawa put it: 'The symbol
is NOT the thing symbolized; the word is NOT the thing; the map
is NOT the territory it stands for.' In the same vein, the model
is NOT the real-world.

The Example of Malthus' Modeling. An example of a simple mathematical model, but one
that has had long-term effects, is in the Malthus lecture: Malthus
assumed that population would grow exponentially while subsistence
would grow at best linearly. From these assumptions, Malthus derived
mathematical consequences and proposed policies to try to prevent,
or at least soften, the consequences.

Below is a schematic for a general mathematical
modeling framework and, following the schematic, what Malthus'
model and his proposal look like in that framework.

Real World: Lots
of stuff going on. The French Revolution a decade ago - Ottoman
Empire in decline - East India Company entrenched in South Asia
- Population growth generally considered good by European intellectuals
- Rapid population growth in the resource-rich United States.

Observation (construction):
Through his lens of experience, goals, and intellect, Malthus
observes or constructs the idea of hard times, of misery and vice.
He'd like to do something about these problems. Lots of stuff
is ignored.

The Model and Its Formulation:

Focus/Variables

Malthus has
to decide on what his focus should be and what aspects of the
real world he should ignore. He picks just two variables to work
with, ignoring all else.

Assumptions

He makes assumptions
about the rates of change of his two variables.

Derivations

He draws conclusions
purely from the mathematics.

Predictions/Comparison to Real World: Now Malthus interprets his mathematical conclusions
in terms of the real world and compares the real world to the
model. Well, ideally, he would do that. But, he doesn't live in
an information-rich age, and he's dealing with lengthy time spans,
so he can't make such comparisons very easily.

Revise Model: If he did this, he didn't tell us about it.

Policy Changes: On the basis of his model, Malthus makes
some recommendations concerning agricultural, labor, and manufacturing
policies, personal restraint, and public assistance policies.

Looking at Malthus' model more closely, we
see the following:

Population

Birthrate - Deathrate = fixed
percent of population per unit time

Food

Agricultural Growthrate = fixed
absolute amount per unit time

Population and Food Supply are both determined
by rates of growth - The rates of growth are unaffected by anything,
except for the modeler's (Malthus') assumptions - a sort of 'invisible
hand' (using the words of Adam Smith, but in a context other than
Smith intended).

What to Focus on: A Critical Choice in
Modeling. An early mathematical
model was formed for the psychology of perception.Things to include
or ignore: Whether the perceiving was going on inside or outside,
if inside, what size the room is, the temperature, noise level,
type of stimulus, distance from the stimulus, length of the stimulus
if it is a card, color of the stimulus card, shortest perceptible
difference in the length of the stimulus card, ... . By focusing
on just two variables, (Magnitude of the stimulus and the least
perceptible difference in the magitudes of the stimuli) the concept
of 'just noticeable difference' was constructed and the Weber-Fechner
Law formulated.

Start Simply.
If you were going to model population growth, what factors or
variables would you want to include? (We'll come back to this
question near the end of class.) Obviously, a lot of potentially
valuable factors related to population growth have been left out.
However, the Malthus model illustrates a principle for beginning
a mathematical model: Start simply - then gradually add complexity,
so long as complexity also adds insight.

To see be more precise about the process of
modeling, including the revision part, let's look at the model
formulation, guided by Lab 1 and the modeling software Stella.

Decide on the variables. In Lab 1, we choose
at first two variables, Population and Births (per year), with
Population represented with the Stella idea of a stock and Births
per year represented with the Stella idea of a flow. Time is also
a variable, but it isn't explicitly controlled in any way. Stella
assumes that all models involve time, and we choose Stella as
our model-creating tool. The initial Population is assumed to
be 1,000, and Births per year is assumed to be 200. The interrelationship
between the two variables is:

Population
at a particular point in time

is

Population
in the previous year + Births

Symbolically, in Stella language this looks
like:

Population
(t) = Population (t-dt) + Births per year * dt
INIT Population = 1000
INFLOWS
Births per year = 200

So, we have a very simple model that we can
compare to reality. How do they compare. Well, if we run this
Stella-implemented model, we see the linear growth of population
over time. What we actually have is a model, supposedly for population
growth, that matches Malthus' model for food growth.

We know that we have at least omitted a very
obvious variable: Deaths! We could make our model a little more
complex by adding a Death variable, represented in Stella as a
flow away from Population.

Population
at a particular point in time

is

Population
in the previous year + Births - Deaths

If Deaths per year is assumed to be 100, we
get symbolically,

Population (t) = Population (t-dt) + Births
per year * dt - Deaths per year * dt
INIT Population = 1000
INFLOWS
Births per year = 200
OUTFLOWS
Deaths per year = 100

The Stella Diagram is

However, the graph of Population over time
is still a linear one, just increasing at a constant absolute
rate that is less than before. Can we do better?

Add Complexity.
A key point to Malthusian growth is the idea of proportional or
percentage growth, but the models thus far are assumed to have
constant absolute growth rates. To introduce percentage growth,
think about the increase in population when that increase is proportional
to the population level itself. In other words, the increase is
dependent upon two things: The proportion and the Population.

Rather than having Births determined by an
'invisible hand,' we'll have it determined by a fixed proportion
and by the Population itself. To do this in Stella, introduce
a converter, called Per Capita Births per year, which will be
constant. Then draw connections from Per Capita Births per year
and from Population to the flow Births per year, so that the diagram
looks like:

Do the same sort of thing with Deaths, making
it dependent upon a constant Per Capita Deaths per year and upon
Population. Now the diagram looks like:

Now if we put in some reasonable numbers for
Per Capita Births per year and Per Capita Deaths per year and
run the model, we get a graph that is an exponential one. How
does this compare to Malthus? How does it compare to our knowledge
of past population growth?

Well, the model pretty well captures Malthus'
ideas about population growth, but it turns out not to fit well
with the data on population growth either on a world-wide basis
or looking at smaller segments, say by country or region of the
world. This isn't surprising, since we haven't taken any real-world
features into account, other than the propensity of populations
to grow!

We'll put in one last factor to try to make
the model more realistic, leaving it to your laboratory work to
take into account food, agricultural innovation, fertility rates,
education, social security systems, etc. This last factor is the
idea of a carrying capacity. It seems reasonable that our earth
and solar system have some limit to the population it can support.
If so, it has a carrying capacity, or maximum number of individuals
that can survive on the planet.

The previous model is extended by adding a
converter called Carrying Capacity, which is a (pretty big) constant.
It is connected to Births per year, so that the diagram is:

The equations for the model are:

Population (t) = Population (t-dt) + Births
per year * dt - Deaths per year * dt
INIT Population = 1000
INFLOWS
Births per year = Per Capita Births per year * Population * (Carrying
Capacity - Population)
OUTFLOWS
Deaths per year = Per Capita Deaths per year * Population

[Note on the term 'Per Capita Births per year' in this model
that includes Carrying Capacity.]

Thus, the number of births per year is assumed
in this model to be jointly proportional to the population and
how close the population is to carrying capacity. When population
is graphed against time it is an elongated, roughly S-Shaped curve
shown below: (Population in billions, time in tens of years)

Does the Model Explain?
Is the graph above consistent with Malthus? Other population data?
What is the significance of the flat growth for the latter portions
of time? The graph does bear some resemblance to the graph distributed
in the Malthus lecture. (You can obtain this graph at the
United Nations Population Information Network.)

Although the graph above is not consistent
with Malthus' assumption about population growth having a constant
doubling time, it perhaps captures some of the spirit in the following
way: There is early on an increasing rate of growth as the graphs
'curves upward.' The fact that the graph increases at smaller
and smaller rates of growth as it approaches a carrying capacity
is possibly what Malthus had in mind by the term 'misery.' This
slow approach to a carrying capacity is perhaps the result of
war, pestilence, and starvation as more and more people contend
for the resources that are now at their upper bound.

What is clear is that even if the graph
were a good depiction of actual world population growth, it doesn't
explain much. The dynamics of population growth remain
a mystery. None of the dynamic interaction of the factors related
to population growth are either assumed in or deduced from the
present model.

So, let us turn to brainstorming other factors
and variables to add to the model.

Brainstorming The Choice of Factors to
Include when Modeling Growth of Populations. If you were going to model population growth to explain
the dynamic interaction of variables involved with growth, what
factors or variables would you want to include? Click below for
student responses:

A Note on the Idea of Parameter.The notion of parameter is inherent to mathematical
modeling. Roughly speaking, the parameters of a model are the
constants involved in the model.

For example, we initially set Births per year
equal to a constant, and we could have set that constant equal
to anything we wanted. For that reason, Births per year was a
parameter in the initial model. Once we changed the model by adding
Per Capita Births per year and set Births per year equal to the
product of Per Capita Births per year and Population, Births per
year was no longer a parameter, but Per Capita Births per year
became a new parameter, which we could set equal to something
like 0.03. Introducing Carrying Capacity into the model introduced
yet another parameter into the model.

Values for the parameters of a model are usually
decided upon by collecting data or experimenting. However, values
may be set in any way the modeler wants and the resulting model
'run' to see what the consequences are. The ability to experiment
in this way is a very useful property of a mathematical model.

The Values of Mathematical Modeling.

1.

One
is forced to choose what to focus on. You must prioritize factors.

2.

The
modeling process helps make thoughts more precise.

3.

A
model helps one go beyond the surface of a phenomenon to an understandingof mechanisms and relationships.

4.

One
can play out different scenarios, modifying assumptions, initial
values, and values of parameters, to see the resulting effects.

Problems Associated with Mathematical
Modeling.

1.

The
model doesn't address what you want to accomplish.

2.

The model is
very sensitive to initial conditions or to the values of parameters.

3.

The
model creates a mathematical solution to a problem that doesn't
lend itself to a mathematical solution.